Questions: Find the horizontal and vertical asymptotes of the curve. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) y=(3x^2+x-1)/(x^2+x-72)

Solution Steps

To find the horizontal and vertical asymptotes of the given rational function, we need to follow these steps:

1. Vertical Asymptotes: Set the denominator equal to zero and solve for \( x \). These values of \( x \) are where the vertical asymptotes occur, provided the numerator is not zero at these points.
2. Horizontal Asymptotes: Compare the degrees of the numerator and the denominator:
   • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
   • If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
   • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (DNE).

To find the vertical asymptotes, we set the denominator equal to zero and solve for \( x \): \[ x^2 + x - 72 = 0 \] Solving this quadratic equation, we get: \[ x = -9 \quad \text{and} \quad x = 8 \] Thus, the vertical asymptotes are at \( x = -9 \) and \( x = 8 \).

To find the horizontal asymptote, we compare the degrees of the numerator and the denominator. Both the numerator \( 3x^2 + x - 1 \) and the denominator \( x^2 + x - 72 \) are of degree 2.

Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients: \[ \frac{3}{1} = 3 \] Thus, the horizontal asymptote is \( y = 3 \).

Final Answer